derivative of e^x(sin(x-cos(x)))

解

\frac{d}{d x} (e^{x} (sin (x) - cos (x)))

2 e^{x} sin (x)

解答ステップ

\frac{d}{d x} (e^{x} (sin (x) - cos (x)))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{x}, g = sin (x) - cos (x)

= \frac{d}{d x} (e^{x}) (sin (x) - cos (x)) + \frac{d}{d x} (sin (x) - cos (x)) e^{x}

\frac{d}{d x} (e^{x}) = e^{x}

\frac{d}{d x} (sin (x) - cos (x)) = cos (x) + sin (x)

= e^{x} (sin (x) - cos (x)) + (cos (x) + sin (x)) e^{x}

簡素化

e^{x} (sin (x) - cos (x)) + (cos (x) + sin (x)) e^{x} : 2 e^{x} sin (x)

= 2 e^{x} sin (x)